How quickly does a chick embryo grow? Remember
that the way any embryo develops is by repeated
cell division. It starts off as a single cell
which spilts into two. Each of those two cells
then splits into two, and so on. We have
illustrated this schematically with a square box
representing the initial egg cell

Move your cursor into the box to `fertilize the
egg' and start the cell divison procedure. The
box divides into 2 regions, then 4, then 8, etc.
The number of regions grows in the same way that
the number of cells in a real embryo does. At
the left, we graph the number of cells against
the number of times cell division has occurred.
Notice how quickly the number of cells
grows!

We can very efficiently describe the way the
number of cells grows if we use some mathematical
notation. Let's denote the number of times
division has occured by the letter t, and
let's denote the total number of cells by
N(t). Thus N(1) denotes the number
of cells after 1 division; so
N(1)=2. Similarly, N(2)=2x2=4,
N(3)=2x2x2=8, etc. A more compact way to
write 2x2x2 is as 23;
2x2x2x2 is 24 and so on.
With this convenient notation, we can write:

N(1) = 21,
N(2) = 22,
N(3) = 23,
N(4) = 24.

Following this pattern, we can write down a
formula for N(t), namely:

N(t)=2t

In mathematical language, N(t) is an
exponential function of t.

In the graph next to the cell division
simulation, we have plotted N(t) for
t=1,2,3,.... We can join up the dots to
get a smooth curve, from which we can read off
values for 2t corresponding to
any value for t, whole number or not. (Later we will see a more
precise way to define 2t for any value of
t.)

In the exponential function N(t)=2t, the number 2
is called the base of the exponential. We
can change this base and get other functions of
t, all of which are called exponentials.
For example, using base 3, we get the
exponential function 3t.

The graphs of exponential functions are
shown in this demonstration. You can
change the base of the exponential
(denoted by b) by using the slider
at the top.

Notice that the graphs extend into the
region where t is negative. This
doesn't make much sense if t
represents the number of times that a
chick egg has divided. Mathematically,
however, we can make sense of it as
follows. For the exponential with base
b, i.e. for the function
bt, we use the
rule that:

b-t
= 1 / bt

This rule tells us how to compute
bt for negative
values of t. With this rule it is
still true that bt+1 = b bt.

If you experiment with a few different choices
for the base, you will notice that the graphs of
all the exponential functions have a few things in
common:

The graphs all pass through the same point
when t=0.

Going off to one direction (left or right),
the value of the function (the height of the
curve) grows very rapidly, but going off to the
other direction, the value decays slowly towards
zero, getting ever closer to the horizontal axis
but never actually reaching it.

Notice that for some choices of base
b, the value increases when we move to
the left and decays to the right, while for
other choices of b, the value increases
to the right and decays to the left. See if
you can figure out what determines which
alternative applies to a given exponential
function.

There is a third, more subtle, property
shared by all the exponential functions. To
understand what this is, we need to look at the
straight lines tangent to the graphs.
In particular, we need to look at the directions
of these tangent lines, as measured by their slopes.

Select a point on the graph by clicking on it.
The demonstration shows the tangent line to the
graph at the selected point. The three numbers
shown at the bottom of the demonstration are:

the value of the
exponential, N = bt, at that
point,

the slope
(denoted m) of the tangent line at that
point, and

the ratio
m/bt

Notice that along a given curve, while the
first two numbers change, the third one remains
the same no matter which point you select! This
is the third special property of
exponentials:

The slope of the tangent line is
proportional to the value of the
exponential at the point of
tangency.

The ratio (the third number displayed in the
applet) is called the constant of
proportionality.

The Special Base

If you explore this property of exponentials in
more detail, you will notice that this constant of
proportionality changes depending on the base of
the exponential. For instance, for the
exponential to base 2 the constant is .693147,
while for the exponential to base 3 the constant
is 1.0986.

This prompts the following question:

Is there some
base for which the constant of proportionality is
exactly 1?

The answer to this question is YES! This base,
for which the exponential function has slope equal
to its value, is traditionally denoted by the
letter e. This number e has many
wonderful properties. You can explore the graph
of the exponential function et, and verify that it
has slope equal to value by clicking the e
button in the demonstration above.